\magnification=\magstep1

$$
P \quad = \quad n \thinspace k \thinspace T
$$

$$
\kappa \quad \propto \quad {{\rho}\over{T^{3.5}}}
$$

$$
\lambda \quad = \quad 4 L 
$$

$$
f \quad = \quad {{v_s}\over{\lambda}} \quad = \quad {{v_s}\over{4 L}}
$$

$$
{\rm period }\ \quad = \quad {{1}\over{f}} \quad = \quad 4 {{L}\over{v_s}}
$$

$$
{\rm period }\ \quad \sim \quad 4 \ \left( {{R}\over{v_s}} \right)
$$

$$
v_s \quad = \quad \sqrt{{T}\over{\mu}}
$$

$$
v_s \quad \sim \quad \sqrt{{P}\over{\rho}}
$$

$$
P \quad \sim \quad {{G M \rho}\over{R}}
$$

$$
v_s \quad \sim \quad \sqrt{{G M}\over{R}}
$$

$$
{\rm period\ } \quad \sim \quad {{4 R}\over{\sqrt{{G M}\over{R}}  }} 
     \quad = \quad 4 \ \left( {{G M}\over{R^3}} \right)^{-1/2}
$$

$$
L \quad = \quad \left( 4 \pi R^2 \right) \ \sigma T^4
$$

$$
L \quad \propto \quad R^2 \qquad \rightarrow \qquad R \quad \propto \quad L^{1/2}
$$

$$
{\rm period \ } \quad \propto \quad M^{-1/2} \ R^{3/2}
$$

$$
{\rm period \ } \quad \propto \quad M^{-1/2} \ L^{3/4}
$$


$$
L \quad \sim \quad M^4 \qquad \rightarrow \qquad M \sim L^{1/4}
$$

$$
{\rm period \ } \quad \propto \quad (L^{1/4})^{-1/2} \ L^{3/4}
  \quad \sim \quad L^{5/8}
$$




\bye
